User:Lucius Chiaraviglio/Musical Mad Science: Difference between revisions
→Musical Mad Science Musings on Diatonicized Sixth-Tone Sub-Chromaticism(?): Add dark generator, 7-limit infill extensions |
→Musical Mad Science Musings on Diatonicized Sixth-Tone Sub-Chromaticism(?): Add commas for 5th harmonic and 7th harmonic infill extensions |
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# As the number of L intervals in a nL 2s scale grows, the range of qualifying generator sizes shrinks, and so the scale becomes more brittle to tempering of the generator, and it becomes hard to find good ratios for specifying the generator. Considering the wider of each pair of generators, the range of [[5L 2s]] (as in [[Meantone]], [[Superpyth]], and their relatives) is very wide range — you have to have a ''bad'' fifth to land outside of its range. The range of [[7L 2s]] is still fairly wide, going from barely over [[52/35]] down to somewhat under [[81/55]]; [[9L 2s]] is narrower, going from barely over [[25/17]] down to somewhat under [[19/13]]; [[11L 2s]] ([[Ivan Wyschnegradsky]]'s original Diatonicized Chromatic scale) brackets [[16/11]]; and the ranges get progressively narrower and the ratios more complicated until by the time we get to 19L 2s, the range falls between two ratios, the second of which is not even all that simple: [[13/9]] and [[36/25]]. The first is too sharp by somewhat over 1{{c}}, and the second is barely too flat; although since it is near-just as 10 steps of [[19edo]], which is equalized 19L 2s, we can count it as snapping to the lower end. It is possible to come up with more complicated ratios by mediation between these slightly out-of-bounds endpoints, such as [[75/52]] and [[49/34]], or even [[62/43]] in the middle, but the latter uses unacceptably large primes, while the previous ratios and even 36/25 itself fail to map properly in the patent [[val]]s of some of the equal temperaments within the range of 17L 2s (this flaw of 36/25 making it tempting to use the slightly flatter [[23/16]], so before considering the next point, it seems better to specify the generator as a tempered 36/25 ~ 13/9, or perhaps even 23/16 ~ 13/9, either way with the proviso that the generator can never reach the just value of either endpoint without going out of range. But the choice of generator tempering comma will need to depend upon which subgroup(s) counts as the core of this temperament, so let's not throw out any of the above intervals just yet. | # As the number of L intervals in a nL 2s scale grows, the range of qualifying generator sizes shrinks, and so the scale becomes more brittle to tempering of the generator, and it becomes hard to find good ratios for specifying the generator. Considering the wider of each pair of generators, the range of [[5L 2s]] (as in [[Meantone]], [[Superpyth]], and their relatives) is very wide range — you have to have a ''bad'' fifth to land outside of its range. The range of [[7L 2s]] is still fairly wide, going from barely over [[52/35]] down to somewhat under [[81/55]]; [[9L 2s]] is narrower, going from barely over [[25/17]] down to somewhat under [[19/13]]; [[11L 2s]] ([[Ivan Wyschnegradsky]]'s original Diatonicized Chromatic scale) brackets [[16/11]]; and the ranges get progressively narrower and the ratios more complicated until by the time we get to 19L 2s, the range falls between two ratios, the second of which is not even all that simple: [[13/9]] and [[36/25]]. The first is too sharp by somewhat over 1{{c}}, and the second is barely too flat; although since it is near-just as 10 steps of [[19edo]], which is equalized 19L 2s, we can count it as snapping to the lower end. It is possible to come up with more complicated ratios by mediation between these slightly out-of-bounds endpoints, such as [[75/52]] and [[49/34]], or even [[62/43]] in the middle, but the latter uses unacceptably large primes, while the previous ratios and even 36/25 itself fail to map properly in the patent [[val]]s of some of the equal temperaments within the range of 17L 2s (this flaw of 36/25 making it tempting to use the slightly flatter [[23/16]], so before considering the next point, it seems better to specify the generator as a tempered 36/25 ~ 13/9, or perhaps even 23/16 ~ 13/9, either way with the proviso that the generator can never reach the just value of either endpoint without going out of range. But the choice of generator tempering comma will need to depend upon which subgroup(s) counts as the core of this temperament, so let's not throw out any of the above intervals just yet. | ||
# In 36edo, the original inspiration for this attempt at a temperament, 19L 2s lends itself to making good use of 36edo as a 2.3.7... subgroup temperament, with the generator 19\36. With this scale, it is possible to choose a mode of this scale (UDP 11|7, cyclic order 14, LLLLLsLLLLLLLLLsLLL, no mode name assigned yet) that includes the following key 2.3.7 intervals: root (0\36), [[9/8]] (6\36), [[7/6]] (8\36), both flavors of split neutral third (10\36 and 11\36), [[9/7]] (13\36), [[4/3]] (15\36), [[3/2]] (21\36), [[7/4]] (29\36), [[16/9]] (30\36), and on to the root, all the while filling in the scale with 2\36 stacked to various extents. It also includes the generator interval 19\36, but let's not assign the generator a (tempered) ratio just yet. The choice of other modes enables use of other intervals relative to the root, while a decent subset of them still support both the 3-limit fourth and fifth. (But see later parts of this analysis, in which it is actually necessary to assign the 2.3.5... subgroup mapping first, at least for the hard half of the tuning spectrum.) | # In 36edo, the original inspiration for this attempt at a temperament, 19L 2s lends itself to making good use of 36edo as a 2.3.7... subgroup temperament, with the generator 19\36. With this scale, it is possible to choose a mode of this scale (UDP 11|7, cyclic order 14, LLLLLsLLLLLLLLLsLLL, no mode name assigned yet) that includes the following key 2.3.7 intervals: root (0\36), [[9/8]] (6\36), [[7/6]] (8\36), both flavors of split neutral third (10\36 and 11\36), [[9/7]] (13\36), [[4/3]] (15\36), [[3/2]] (21\36), [[7/4]] (29\36), [[16/9]] (30\36), and on to the root, all the while filling in the scale with 2\36 stacked to various extents. It also includes the generator interval 19\36, but let's not assign the generator a (tempered) ratio just yet. The choice of other modes enables use of other intervals relative to the root, while a decent subset of them still support both the 3-limit fourth and fifth. (But see later parts of this analysis, in which it is actually necessary to assign the 2.3.5... subgroup mapping first, at least for the hard half of the tuning spectrum.) | ||
# It is noteworthy | # It is noteworthy that harmonics 3 and 23 are very stable over the tuning spectrum of this scale (at least for EDO values up into the mid double digits, except for needing a wart at 112b), although the 23rd harmonic is guaranteed to be sharp, meaning that at larger EDO values, increasingly fine divisions of the octave will cause the mapping to disagree with 10\19 and 9\17 (and thereby with 19\36), thus requiring an 'i' [[wart]]. The 7th harmonic is also reasonably stable, although it changes enough over the tuning spectrum to get rather bad at the extremes; the 5th harmonic is definitely not stable in the hard half of the spectrum, but is fairly stable in the soft half (although warts are needed for a few of the larger EDOs). | ||
# Tried assigning the generator as 23/16 ~ 13/9, tempering out [[208/207]] (the vicetone comma). But the problem is that — as can be seen in the table of harmonics below — the 13th harmonic is not stable enough for the entire 17L 2s tuning spectrum, even for the for the hard half of the tuning spectrum (closer to just 13/9, including having the best 3rd harmonic within the tuning spectrum). Need to split the tuning spectrum of 17L 2s into 2 or more temperaments. Still have not found the right mix of harmonics for the hard half (36edo and harder), but for the soft half, the 5th and 53rd harmonics are stable enough to team up with the 3rd and 23rd harmonics to get a usable generator. Nobody in their right mind is going to want to actually use the 53rd harmonic for constructing intervals, but for tuning the generator, it will have to do. The bright generator (basic 19\36, spectrum from 10\19 soft to 9\17 hard) is therefore constituted as [[23/16]] (|-4 0 0 0 0 0 0 0 1⟩, 628.274347{{c}}) ~ [[384/265]] (|7 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1⟩, 642.1367415{{c}}), tempered together using the unnamed comma [[6144/6095]] (|11 1 -1 0 0 0 0 0 -1 0 0 0 0 0 0 -1⟩, 13.8623942563{{c}}); this comma is in fact tempered out in most of the EDOs on the soft half of the tuning spectrum, plus 17edo constituted as the often-used (and barely further from just) 17c val. It follows that the dark generator is constituted as [[32/23]] (|5 0 0 0 0 0 0 0 -1⟩, 571.725653{{c}}) ~ [[265/192]] (|-6 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 1⟩, 557.863258{{c}}), tempered together using the same comma. (Coming in the future: More work on the hard half of the tuning spectrum. Maybe the 7th and 17th harmonics are stable enough for the middle of the 17L 2s tuning spectrum?) | # Tried assigning the generator as 23/16 ~ 13/9, tempering out [[208/207]] (the vicetone comma). But the problem is that — as can be seen in the table of harmonics below — the 13th harmonic is not stable enough for the entire 17L 2s tuning spectrum, even for the for the hard half of the tuning spectrum (closer to just 13/9, including having the best 3rd harmonic within the tuning spectrum). Need to split the tuning spectrum of 17L 2s into 2 or more temperaments. Still have not found the right mix of harmonics for the hard half (36edo and harder), but for the soft half, the 5th and 53rd harmonics are stable enough to team up with the 3rd and 23rd harmonics to get a usable generator. Nobody in their right mind is going to want to actually use the 53rd harmonic for constructing intervals, but for tuning the generator, it will have to do. The bright generator (basic 19\36, spectrum from 10\19 soft to 9\17 hard) is therefore constituted as [[23/16]] (|-4 0 0 0 0 0 0 0 1⟩, 628.274347{{c}}) ~ [[384/265]] (|7 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1⟩, 642.1367415{{c}}), tempered together using the unnamed comma [[6144/6095]] (|11 1 -1 0 0 0 0 0 -1 0 0 0 0 0 0 -1⟩, 13.8623942563{{c}}); this comma is in fact tempered out in most of the EDOs on the soft half of the tuning spectrum, plus 17edo constituted as the often-used (and barely further from just) 17c val. It follows that the dark generator is constituted as [[32/23]] (|5 0 0 0 0 0 0 0 -1⟩, 571.725653{{c}}) ~ [[265/192]] (|-6 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 1⟩, 557.863258{{c}}), tempered together using the same comma. (Coming in the future: More work on the hard half of the tuning spectrum. Maybe the 7th and 17th harmonics are stable enough for the middle of the 17L 2s tuning spectrum?) | ||
# To get ~[[3/1]], we stack 3 bright generators (and then octave reduction gets us ~[[3/2]]), putting this temperament on the alpha-tricot part of the [[ploidacot]] system (not to be confused with the [[Alphatricot]] temperament, and not to be confused with [[Mothra]], which is tricot without the alpha, which only works for the subset of the following EDOs that also have octaves divisible by 3). The comma for this is [[12288/12167]] (|12 1 0 0 0 0 0 0 -3⟩, 17.13195906{{c}}) in the 23-limit, and [[18874368/18609625]] (|21 2 -3 0 0 0 0 0 0 0 0 0 0 0 0 -3⟩, 24.45522370876{{c}}) in the 53-limit. Except as noted with warts, this works for the patent vals of all of the smaller and mid-size EDOs in the 17L 2s tuning spectrum (leaving out the far right column of the tuning spectrum table apart from its top and bottom ends, since the EDO sizes in between the top and bottom ends get very large). As noted above, things are different between the soft half (up to and including 36edo) and the hard half (36edo onwards) later on, so even though they work the same way for the 3rd harmonic, I split the EDO list in half, although oddly enough 17c works as if it was on the soft half despite being at the hard end (collapsed 17L 2s). EDO list for soft half of tuning spectrum: 19, 36, 55, 74, 91c, 93, 112b, 127cci, 129, 146i (and note that 112b just barely misses being the patent val of the dual-fifth 112edo). EDO list for hard half of tuning spectrum: 17, 36, 53, 70, 87, 89, 104, 123, 125, 142 (this list is manually generated — [http://x31eq.com/temper/ Graham Breed's x31eq temperament finder] has trouble with the high prime limit of the subgroup, and only finds part of the spectrum, and shows some other EDOs in addition). | # To get ~[[3/1]], we stack 3 bright generators (and then octave reduction gets us ~[[3/2]]), putting this temperament on the alpha-tricot part of the [[ploidacot]] system (not to be confused with the [[Alphatricot]] temperament, and not to be confused with [[Mothra]], which is tricot without the alpha, which only works for the subset of the following EDOs that also have octaves divisible by 3). The comma for this is [[12288/12167]] (|12 1 0 0 0 0 0 0 -3⟩, 17.13195906{{c}}) in the 23-limit, and [[18874368/18609625]] (|21 2 -3 0 0 0 0 0 0 0 0 0 0 0 0 -3⟩, 24.45522370876{{c}}) in the 53-limit. Except as noted with warts, this works for the patent vals of all of the smaller and mid-size EDOs in the 17L 2s tuning spectrum (leaving out the far right column of the tuning spectrum table apart from its top and bottom ends, since the EDO sizes in between the top and bottom ends get very large). As noted above, things are different between the soft half (up to and including 36edo) and the hard half (36edo onwards) later on, so even though they work the same way for the 3rd harmonic, I split the EDO list in half, although oddly enough 17c works as if it was on the soft half despite being at the hard end (collapsed 17L 2s). EDO list for soft half of tuning spectrum: 19, 36, 55, 74, 91c, 93, 112b, 127cci, 129, 146i (and note that 112b just barely misses being the patent val of the dual-fifth 112edo). EDO list for hard half of tuning spectrum: 17, 36, 53, 70, 87, 89, 104, 123, 125, 142 (this list is manually generated — [http://x31eq.com/temper/ Graham Breed's x31eq temperament finder] has trouble with the high prime limit of the subgroup, and only finds part of the spectrum, and shows some other EDOs in addition). Note that some of the commas in each comma spectrum listed below (corresponding to the generator spectrum above) have negative just intonation values, because each spectrum crosses through 0. | ||
# Dividing the EDO list into hard and soft halves is helpful for looking at these EDOs as 5-limit temperaments. For the soft half, except as noted with warts, these are all [[Meantone]] temperaments, or conntortions under [[Meantone]] temperaments: 19, 36 (contorted under 12), 55, 74, 91c, 93 (contorted under 31), 112b, 127cc, 129 (contorted under 43), 146c. Without the warts, 91edo and 127edo fall on [[Schismic–Pythagorean_equivalence_continuum#Python|Python]] (currently still named Lalagu by Graham Breed's x31eq temperament finder), for which 16 fifths (octave-reduced) are needed to reach [[5/4]]; while 146edo falls on the currently unnamed diploid temperament that flattens the fifth by (optimally) close to 1/10 of |-27 20 -2⟩ to get ~5/4. For the hard half, the picture is more complicated (and the generator constitution is not even currently set), so we'll leave that for later. | # Dividing the EDO list into hard and soft halves is helpful for looking at these EDOs as 5-limit temperaments. For the soft half, except as noted with warts, these are all [[Meantone]] temperaments, or conntortions under [[Meantone]] temperaments: 19, 36 (contorted under 12), 55, 74, 91c, 93 (contorted under 31), 112b, 127cc, 129 (contorted under 43), 146c. Without the warts, 91edo and 127edo fall on [[Schismic–Pythagorean_equivalence_continuum#Python|Python]] (currently still named Lalagu by Graham Breed's x31eq temperament finder), for which 16 fifths (octave-reduced) are needed to reach [[5/4]]; while 146edo falls on the currently unnamed diploid temperament that flattens the fifth by (optimally) close to 1/10 of |-27 20 -2⟩ to get ~5/4. For the hard half, the picture is more complicated (and the generator constitution is not even currently set), so we'll leave that for later. | ||
# For the Meantone set, since 4 fifths get the ~5/4, and 3 (unnamed-53-limit-comma-tempered) bright generators get the fifth, this means that 12 of these bright generators get the ~5/4. (Originally this was going to be a 2.3.5.13.23 subgroup extension of Meantone, with the Vicetone comma already having a fitting name, leading "Vicetone" as the name for this extension; but the 13th harmonic mapping just wasn't stable enough. Too bad. For now, going to have to go with something weird like Fitho-vicesimotertial for the comma and temperament name.) Dropping EDOs for which warts greatly degrade accuracy (91c, 127cci, and 142ci), the optimal ET sequence is: 17c, 19, 36, 55, 74, 93, 112b, 129 (this is manually generated — see above about x31eq having trouble with the high prime limit of the subgroup.) | # For the Meantone set, since 4 fifths get the ~5/4, and 3 (unnamed-53-limit-comma-tempered) bright generators get the fifth, this means that 12 of these bright generators get the ~5/4. (Originally this was going to be a 2.3.5.13.23 subgroup extension of Meantone, with the Vicetone comma already having a fitting name, leading "Vicetone" as the name for this extension; but the 13th harmonic mapping just wasn't stable enough. Too bad. For now, going to have to go with something weird like Fitho-vicesimotertial for the comma and temperament name.) Dropping EDOs for which warts greatly degrade accuracy (91c, 127cci, and 142ci), the optimal ET sequence is: 17c, 19, 36, 55, 74, 93, 112b, 129 (this is manually generated — see above about x31eq having trouble with the high prime limit of the subgroup.) The comma for this is made from the [[81/80|syntonic comma]] by substituting each instance of ~3/2 with an octave-reduced stack of 3 of our bright generator, which produces a spectrum of commas from |52 0 1 0 0 0 0 0 -12⟩ ~ |-80 -12 13 0 0 0 0 0 0 0 0 0 0 0 0 12⟩, of which |19 -3 4 0 0 0 0 0 -9 0 0 0 0 0 0 3⟩ (made by substituting 9 instances of (3/2)<sup>(1/3)</sup> by 23/16 and the other 3 instances of (3/2)<sup>(1/3)</sup> by 384/265) has the closest 53-limit just intonation value to 0 (5.4343638749{{c}}). Naturally, 81/80 itself is also tempered out. | ||
# It is natural to ask next for the 7-limit infill extension. | # It is natural to ask next for the 7-limit infill extension. | ||
## For some of these EDOs the normal [[Septimal Meantone]] extension gives the proper ~[[7/4]], only having the generator number multiplied by 3 since 3 bright generators are needed to get 1 fifth — optimal ET sequence: 19, 55d, 74, 93, 112b; of these, only 93 qualifies for Mothra. The comma for this is [[Harrison's comma]] with instance of 3/2 substituted by on octave-reduced stack of 3 of our bright generator, which produces a spectrum of enormously complicated commas from |-133 0 0 -1 0 0 0 0 30⟩ ~ |197 30 -30 -1 0 0 0 0 0 0 0 0 0 0 0 -30⟩, of which |-34 9 -9 -1 0 0 0 0 21 0 0 0 0 0 0 -9⟩ (made by substituting 21 of 30 instances of (3/2)<sup>(1/3)</sup> by 23/16 and the other 9 of 30 instances of (3/2)<sup>(1/3)</sup> by 384/265) has the closest 53-limit just intonation value to 0 (4.16605989{{c}}), while still clocking in at 66 digits. | |||
## For those EDOs having a less flat fifth (but also including 19), the extension is actually much simpler, needing only 11 bright generators to get ~[[7/4]] — optimal ET sequence: 17c, 19, 36, 55 of these, only 36 qualifies for Mothra. The comma for this is a spectrum of less complicated (but still very complicated) commas from |47 0 0 1 0 0 0 0 -11⟩ ~ |-74 -11 11 1 0 0 0 0 0 0 0 0 0 0 0 11⟩, of which |3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩ has the closest 53-limit just intonation value to 0 (2.35850949135{{c}}, made using 7 instances of 23/16 and 4 instances of 384/265). | |||
## This leaves out 129edo, which we don't want to miss because it has a very accurate 7th harmonic; for 129edo, if we want a strong extension, we need -44 bright generators (which is +44 dark generators) — optimal ET sequence: 55, 74, 129; however, this is overly complex for all 3 members, since 55 and 74 also belong to much simpler strong extensions, while 129 qualifies for Mothra. For 129edo, this means that we can proceed by -3 bright generators (+3 dark generators), octave-reduce, and divide by 3, which simplifies to +1 dark generator and +1/3 octave; furthermore, this also works for the other EDO sizes divisible by 3, which suggests the name Alpha-Mothra (since these are both tricot and alpha-tricot, which simplifies to triploid alpha-dark_generator); optimal ET sequence: 36, 93, 129. The comma for the simplified form is a spectrum of merely highly complicated commas from [[4173281/4194304]] (|3 -4 4 1 0 0 0 0 -7 0 0 0 0 0 0 4⟩) ~ [[18966528/18609625]] (|11 3 -3 3 0 0 0 0 0 0 0 0 0 0 0 -3⟩), of which [[544341/542720]] (|-11 1 -1 3 0 0 0 0 2 0 0 0 0 0 0 -1⟩) has the closest 53-limit just intonation value to 0 (5.1631554689{{c}}, smaller than the [[1029/1024|Gamelisma]] for which it substitutes). | |||
Added: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 08:20, 4 April 2025 (UTC)<br> | Added: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 08:20, 4 April 2025 (UTC)<br> | ||
Last modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) | Last modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 09:41, 23 April 2025 (UTC) | ||
=== Table of odd harmonics for various EDO values supporting 17L 2s === | === Table of odd harmonics for various EDO values supporting 17L 2s === | ||